Horizontal rolls over localized heat source in a cylindrical layer

Highlights

• Convection over a localized heat source was studied experimentally and numerically.

• Complex system of secondary flows over the heater was observed.

• Transitions from transverse rolls to radial rolls were found.

• Transverse rolls formation is a periodic process within a wide range of parameters.

Abstract

Convection over a localized heat source in a cylindrical layer was studied experimentally and numerically for fluids with different values of Prandtl number. A basic flow produced by a horizontal temperature gradient occupies the whole layer and leads to unstable temperature stratification over the heating area and the formation of a complex system of secondary flows. The main focus of our study was the spatial and temporal evolution of small-scale convective structures in the boundary layer of a basic flow. Transitions from transverse rolls to radial rolls and further to their superposition were found in experiment and numerical simulation. Various types of visualization revealed co-existence of different kinds of secondary flows. Complex convective patterns over the heating area are temporally periodic. The characteristic frequency of transverse rolls depends on Rayleigh number for a wide range of governing parameters.

Introduction

Secondary flows in the form of horizontal rolls are a common feature of a large variety of flows of different nature and scale. The first studies of horizontal rolls in advective flow were done in connection with stability analysis of Hadley circulation, ocean circulations and crystal growth  [1], [2]. A thin layer of fluid with an applied horizontal temperature gradient and no lateral boundaries was considered. It was found that both a hydrodynamic mechanism (vortex excitation on the boundary of counter-streaming flows) and a convective mechanism (Rayleigh instability in boundary layers characterized by unstable temperature stratification) may lead to advective flow instability. Convective instability in the boundary layer can generate two different modes in the form of transverse and longitudinal rolls, but the longitudinal rolls remain the dominant mode for all values of Prandtl number  [2]. Interactions between atmospheric flow and land or sea often lead to the horizontal rolls in the planetary boundary layer  [3]. Depending on their size and strength, rolls play a significant role in transporting momentum, heat and moisture through the atmospheric boundary layer. Horizontal convective rolls known as cloud streets can be clearly seen on the satellite images. The first evidence for the formation of horizontal rolls oriented along the mean flow in the boundary layer of tropical cyclones was presented in  [4]. Recent full-scale studies strongly suggest that the rolls are a typical structure of meso-scale convective systems  [5], [6], [7] and may have a considerable effect on the heat and mass transfer between the water and the air. A theoretical description of horizontal roll formation in the atmospheric boundary layer was proposed by Foster  [8].

Experimentally, longitudinal rolls were first detected in the convective boundary layer over a heated inclined plate  [9]. The size and intensity of longitudinal rolls increase along the flow. The rolls deform the boundary layer, forming upward and downward flows in the fluid. The increase of vertical velocity in the rolls leads to a laminar–turbulent transition and the breaking down of the rolls. These structures essentially affect the local heat transfer during the laminar–turbulent transition  [10], [11]. In practical applications, horizontal rolls appear in mixed (forced plus natural) convection in channels and they enhance heat transfer in heat exchangers, chemical vapor deposition, cooling systems for electronic equipment and nuclear reactors  [12], [13], [14]. The strong dependence of heat transfer on roll structure has motivated a number of experimental and theoretical studies of mixed convection in channels. In a forced flow in a plane horizontal channel heated from below, the formation of horizontal rolls due to convective instability and the transition of transverse rolls to longitudinal rolls depend on the intensity of the throughflow, characterized by the Reynolds number  [12], [14], [15]. The formation of longitudinal rolls in the thermal entrance region of Poiseuille water flow heated from below with uniform heat flux was studied experimentally by  [12], [13], [16]. The effect of secondary flows caused by natural convection on the laminar–turbulent hydrodynamic transition was studied in  [17]. Horizontal rolls, generated in convective flow above a partially heated bottom in a rectangular box were studied experimentally for a wide range of parameters (the Prandtl number, the Rayleigh number and the aspect ratio)  [18], [19]. A variety of regimes with longitudinal rolls, with transverse rolls and with mixed structures has been observed. It was shown that the structure of secondary flows is defined by the level of convective supercriticality in the boundary layer (Rayleigh number) and the intensity of the throughflow, defined by the Reynolds number, which depends itself on the heating and size, i.e. on the Rayleigh number. Most of the regimes studied were characterized by the appearance of longitudinal rolls. Transverse rolls appeared in the flow only with a large vertical drop in the temperature and weak large-scale flow (which is possible only at large values of the Prandtl number).

Advective flow over a heated plate in rectangular enclosures and its instability, in the form of horizontal rolls, is only a particular case of a convective flow initiated by non-homogeneous heating. Another interesting case is multi-scale convection produced by a circular localized heat source, which is a widespread phenomenon in nature and various technological processes. Convection from a discrete heat source is quite specific because local heating induces both vertical and horizontal temperature gradients. As a result of the applied horizontal temperature gradient, a large-scale convective circulation appears  [20]. When the heater diameter substantially exceeds the layer depth, large-scale circulation leads to the formation of boundary layers with potentially unstable temperature stratification on the periphery of the heating area, where relatively cold fluid flows over a hot surface and creates favorable conditions for the generation of convective rolls  [21]. A survey of studies focused on convection from local heat sources with a cylindrical geometry showed that the studies mostly consider to the formation of large-scale circulation, often turbulent, produced by a discrete heat source without paying close attention to the formation of secondary flows in the boundary layer over the heating area. The structure of convective flow from a small hot spot on the bottom was studied by Torrance  [20], [22]. It was shown that a discrete heat source leads to the formation of a large-scale convective cell. A number of studies have considered convection from so-called urban heat islands  [23], [24], [25], investigating experimentally and numerically large-scale convective circulation from extended local heat sources with ambient temperature stratification. Differential rotation from localized heat source in the center or on the periphery of a cylindrical layer was studied in  [26], [27]. For engineering applications, such as heat exchangers, cooling of modern electronic equipment and nuclear reactors, natural convection in a two-dimensional, rectangular enclosure with localized heating from below and symmetrical cooling from the sides has been simulated  [28]. Most of the studies of small-scale convective patterns in cylindrical enclosures have considered Rayleigh–Benard convection in thin fluid layers with uniform heating from below and uniform cooling from above  [29], [30], [31], [32], [33]. It has been shown that convective patterns may have various shapes, from ideal straight rolls to giant spirals, traveling waves and complex spatial defects. These are fully summarized in a review paper by Bodenschatz, Pesch, and Ahlers  [34]. It should be noted that even for the same values of Raleigh number and aspect ratio, a rich variety of convective patterns can be achieved with different initial conditions  [35]. Convective patterns in thin fluid layers with an applied vertical temperature gradient usually appears without or very weak  [36] large-scale flow. Benard–Marangoni convection in a cylindrical tank with local heating was considered in  [37]. The diameter of the tank was 100 mm, the upper surface was free and the depth of the layer was changed up to 8 mm. A transition from the steady state to a regime when basic flow was superimposed with traveling waves was found. Stability of a thermocapillary flow from a concentrated source of heat located near the free surface was studied in  [38]. Surface waves of different shapes (circle and spiral) were observed. Recent results concerning convective structures in a Benard–Marangoni system with non-homogeneous heating are presented in  [39].

Finally, after reviewing studies of convection in cylindrical enclosures, we may conclude that the formation of secondary flows in the boundary layer of large-scale advective flow over an extended heater has escaped close attention. The first observations of transverse and longitudinal rolls in such a system were presented in  [21]. A similar problem but for small values of Prandtl number was considered in  [40] where instabilities in a fluid layer with a free surface in a cylindrical container non-homogeneously heated from below were studied. It was found that the basic state may bifurcate to different solutions depending on vertical and lateral temperature gradients and on the shape of the heating function. In  [41] a convective instability was described as the mechanism for generating vertical vortices in a cylindrical annulus non-homogeneously heated from below.

In the current study, like  [21], we consider convective system with a focus on the boundary layer dynamics and the formation of small-scale convective structures using a modern experimental technique and numerical simulation. The paper is organized as follows. In Sections  2 Experimental setup, 3 Experimental measurements we describe the experimental setup and the results of experimental measurements. In Section  4 we present the results of the numerical simulation. The formation and specifics of the observed patterns are discussed in detail in Section  5.